Issue 
A&A
Volume 510, February 2010



Article Number  A78  
Number of page(s)  7  
Section  Galactic structure, stellar clusters, and populations  
DOI  https://doi.org/10.1051/00046361/200912886  
Published online  16 February 2010 
Cluster radius and sampling radius in the determination of cluster membership probabilities
N. Sánchez  B. Vicente  E. J. Alfaro
Instituto de Astrofísica de Andalucía, CSIC, Apdo. 3004, 18080 Granada, Spain
Received 14 July 2009 / Accepted 24 November 2009
Abstract
We analyze the dependence of the membership probabilities
obtained from kinematical variables on the radius of the field of view around open clusters (the sampling radius, ). From simulated data, we show that optimal discrimination between cluster members and nonmembers
is achieved when the sampling radius is very close to the cluster radius. At higher
values, more field stars tend to be erroneously assigned as cluster
members. From real data of two open clusters (NGC 2323 and
NGC 2311), we infer that the number of identified cluster
members always increases with increasing .
However, there is a threshold value
above which the identified cluster
members are severely contaminated by field stars and the effectiveness
of membership determination is relatively small. This optimal sampling
radius is 14 arcmin for NGC 2323 and 13 arcmin
for NGC 2311. We discuss the reasons for this behavior and
the relationship between cluster radius and optimal sampling radius. We
suggest that, independently of the method used to estimate membership
probabilities, several tests using different sampling radius should be
performed to evaluate possible biases.
Key words: methods: data analysis  open clusters and associations: general  open clusters and associations: individual: NGC 2311  open clusters and associations: individual: NGC 2323
1 Introduction
Large astrometric catalogues derived from surveys covering very wide areas of the sky are allowing the systematic searching of new star systems (see, for example, Myullyari et al. 2003; LópezCorredoira et al. 1998; Hoogerwerf & Aguilar 1999; Caballero & Dinis 2008; Kazakevich & Orlov 2002; Zhao et al. 2009, and references therein). The searching process is based on the detection of clearly defined structures in subsets of phase space. Both spatial density peaks and proper motion peaks are indicative of star clusters; peaks detectable in only the proper motion distributions suggest the existence of moving groups, whereas more spreadout and less dense velocityposition correlated structures could be associated with stellar streams. Once these structures have been detected, the next step is to identify possible members of the star system. In the particular case of open clusters, the most often used procedure for selecting possible cluster members is the algorithm designed by Sanders (1971). This algorithm is based on a former model proposed by Vasilevskis et al. (1958) for the proper motion distribution. The model assumes that cluster members and field stars are distributed according to circular and elliptical bivariate normal distributions, respectively. The Sanders' algorithm, or some variation or refinement of it, has been and is still being widely used to estimate cluster memberships either as the only method or as part of a more complete treatment that includes, for example, spatial and/or photometric criteria. Some representative references are Wu et al. (2002), Jilinski et al. (2003), BalaguerNúñez et al. (2004), Dias et al. (2006), Kraus & Hillenbrand (2007), and Wiramihardja et al. (2009).
With the advent of large catalogues and databases available via internet and future surveys such as the forthcoming Gaia mission of ESA, the interest in developing and applying fully automated techniques is of increasing interest among the astronomical community. However, special care must be taken to avoid obtaining biased results. In this work, we show that the results obtained when using the Sanders' algorithm depend significantly on the choice of the size of the field of view surrounding the cluster. So, once a possible open cluster is detected, it is natural to ask which area of the sky should be sampled to obtain the most reliable membership determinations. It is equally important to ask about the robustness of used methodology, i.e., how the solution changes when the sampled area is varied? Here we explore these subjects by using both simulated and real data. In Sect. 2, we briefly present the method used to determine memberships and describe the simulations that we performed to analyze the expected behavior. The results of applying the Sanders' algorithm on the simulated data are discussed in Sect. 3. After this, in Sect. 4 we use real astrometric data of two open clusters (NGC 2323 and NGC 2311) to evaluate the performance of the algorithm. We discuss strategies to estimate the optimal sampling radius, i.e., the maximum radius beyond which the identified cluster members are expected to be severely contaminated by field stars. The main results of the present work are summarized in Sect. 5.
2 Description of the method
2.1 Membership determination
The key point of the membership discrimination method
is the assumption that the distribution of observed
proper motions (,
)
can be described
by means of two bivariate normal distributions,
one circular for the cluster and one elliptical
for the field (Vasilevskis et al. 1958). We define
and
to be the cluster and field probability
density functions, respectively. Then,
(1) 
and
=  (2)  
where is the cluster distribution centroid with standard deviation , is the field centroid with standard deviations and , and is the correlation coefficient of field stars. The probability density function for the whole sample is simply
(3) 
and being the normalized numbers of cluster and field stars, respectively. To obtain the unknown parameters (centroids, standard deviations, numbers of members and nonmembers), an iterative procedure is used by applying the maximum likelihood principle (Sanders 1971). Here we use the algorithm proposed by CabreraCaño & Alfaro (1985), which first detects and removes outliers that can produce unrealistic solutions, and then uses a more robust and efficient iterative procedure for the model parameter estimation. Once these parameters are known, the membership probability of the ith stars can be calculated directly as
(4) 
2.2 Simulations
We consider a cluster with a given radius . We define ``cluster radius'' as the radius of the smallest circle that can completely enclose its stars. In true situations, is an unknown quantity that has to be estimated a posteriori, but here its value is known and remains constant throughout each simulation. The total number of stars belonging to the cluster is denoted by and the number of field stars lying exactly within the same sky area of the cluster is . The independent variable is the radius of the field encircling the cluster. This radius might represent the radius of the field in which the observations are made or the field around the cluster extracted from an astrometric catalogue. We call this variable the sampling radius , which can be larger or smaller than the cluster radius .
The numbers of cluster stars and field stars to
be simulated are represented by
and
,
respectively. Obviously, the number
of clusters stars and field stars
within the field of view depend on the
size of this field, that is, both
and
are functions of .
If the
field stars are distributed nearly uniformly in space,
then
should increase as the sampling
radius increases as
The rate at which increases with depends instead on the radial profile of the surface density of cluster stars ( ). For simplicity, we assume that the surface density at r is given by (Caballero 2008)
where the index . For the extreme case , we have . Integrating Eq. (6), we obtain the number of cluster stars within a given sampling radius (for ),
Negative values make no sense, so this approach is limited to the range . The role of the parameter is to control how rapidly increases as increases. Thus, we do not need to know the exact functional form as long as we are able to simulate either completely flat ( ) or extremely peaked ( ) density profiles.
To perform the simulations, we distribute field stars and cluster stars according to bivariate Gaussian distributions in the proper motion space . The routine ``gasdev'' from the Numerical Recipes package (Press et al. 1992) is used to generate normally distributed random numbers. The fields are centered on (0,0) with standard deviations of . The tests performed using elliptical (rather than circular) distributions for the field stars yielded essentially the same results and trends. The clusters are centered on and have standard deviations . Thus, for a given sampling radius and according to Eq. (5), we randomly generate field stars that follow a bivariate normal distribution in the proper motion space. For the cluster, we generate stars according to Eq. (7) when , and we generate stars when . The three free parameters, excluding those describing the Gaussians, are the total number of stars in the cluster ( ), the number of field stars within the cluster area ( ), and the cluster star density profile (). For each set of parameters, we performed 100 simulations and calculated both the average values of the studied quantities and their corresponding standard deviations.
3 Results from simulations
For each simulation, we calculated cluster membership probabilities using the method described in Sect. 2.1. We performed several simulations by varying the input parameters (the number of stars in both the cluster and the field, the centroid distance in the proper motion space, and standard deviations) within reasonable ranges. Apart from minor differences, such as the error bars being larger when cluster and field distributions are more similar, all the results and trends remained essentially identical to those described in this section. We begin by showing how the algorithm works. In Fig. 1, we
Figure 1: Proper motion for the stars of a random simulation with , , and (see text for details of the meaning of each of these quantities). Left panel shows the distribution for all the 442 simulated stars. Red circles are the field stars centered on (0,0) with and blue circles are the 200 cluster stars centered on (1,0)with . Right panel is a magnification of the central region in which we have marked with circles the stars whose resulting cluster membership probabilities are higher than 0.5 according to the algorithm used. 

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What would happen if we select a larger field? To address this point, we calculated membership probabilities as a function of the sampling radius. Here we consider as cluster members stars with membership probabilities 0.5 in a Bayesian sense. We performed several tests of different selection criteria. As expected, the number of assigned members depends on the selection criterion used, although the main results and trends presented here remain unchanged. Figure 2 shows
Figure 2: Calculated number of field and cluster stars as a function of the sampling radius in units of the cluster radius, , for simulations with the same set of parameters as Fig. 1. a) Simulation with peaked density profile ( ), assigned members are indicated by squares connected by lines. b) Simulation with flat density profile (), members are indicated by circles connected by lines. Assigned field stars are indicated by vertical bars connected by lines, the length of the bars indicating one standard deviation. The real numbers of simulated stars are shown by dashed lines (cluster) and dotted lines (field). 

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Figure 3: Calculated fraction of cluster stars as a function of the sampling radius for the same simulations as in Fig. 2. The real (simulated) values are shown by dashed lines. 

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Figures 2 and 3 show the number of stars classified as members, although we do not know whether this classification is indeed reliable. To quantify the correctness of the result, we define the matching fraction of the cluster to be the net proportion of cluster stars that are well classified. If is the total number of cluster stars correctly classified as members minus the number of cluster stars incorrectly classified as nonmembers, then . The value of can be a negative number if the number of misclassifications is higher than the number of correct classifications and is exactly 1 only when the algorithm classifies correctly all the stars of the cluster. In Fig. 4, we
Figure 4: Matching fraction of the cluster (see text) as a function of the sampling radius for the same simulations as in Fig. 2. The error bars are of the order of the symbol sizes but are not shown for clarity. 

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4 Results using real data
We use the CdCSF Catalogue (Vicente et al. 2010), an astrometric catalogue with a mean precision in proper motion of 2.0 mas/yr (1.2 mas/yr for reliable measurements, typically for stars with V < 14). Given the position of a known open cluster, we extract circular fields of varying radius centered on it and then we calculate membership probabilities by using the same algorithm as in Sect. 3. Here we analyze two open clusters that are included in the area covered by this catalogue: NGC 2323 (M 50) and NGC 2311. To minimize the influence of possible outliers on our results, we restrict the sample to mas/yr. The number of probable members , i.e., stars with membership probabilities higher than 0.5, is shown in Fig. 5 as
Figure 5: Number of cluster stars as a function of the sampling radius in arcmin for the open clusters NGC 2323 (squares connected by lines) and NGC 2311 (circles connected by lines). 

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Figure 6: Fraction of cluster stars as a function of the sampling radius for NGC 2323 (squares connected by lines) and NGC 2311 (circles connected by lines). Vertical arrows indicate the optimal sampling radii (see text). 

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The main reason behind the behavior observed in Fig. 6 is the disagreement between the assumed and the ``true'' underlying distributions of proper motion of field stars. A circular normal bivariate function is a good representation of the cluster probability density function (PDF), the standard deviation being the result of observational errors that prevent the intrinsic velocity dispersion of the cluster from being completely resolved. However, it is known that an elliptical normal bivariate function is not always the most reliable model for the field PDF (see discussions on this subject in CabreraCaño & Alfaro 1990; BalaguerNúñez et al. 2004; Uribe & Brieva 1994; Sánchez & Alfaro 2009; Griv et al. 2009). The combination of several factors, such as galactic differential rotation or peculiar motions, may affect the field star distribution, which usually tends to exhibit nonGaussian tails. Nonparametric models, which make no a priori assumptions about the cluster or field star distributions, were introduced and used to overcome this problem (cf. CabreraCaño & Alfaro 1990; Chen et al. 1997). We note that both the classical parametric and nonparametric methods agree reasonably well with each other only for nearly Gaussian field distributions (see Fig. 5 in Sánchez & Alfaro 2009). When the number of field stars increases and the algorithm tries to fit a Gaussian function to the PDF, the fit tends to produce a wider and flatter function. As a consequence, the membership probabilities (defined as the ratio of the cluster to the total proper motion distribution function) increases and the number of assigned members therefore also increases. This effect is magnified when the cluster distribution becomes ``contaminated'' by many field stars, because the standard deviation of the cluster then tends to increase with the consequent increase in the number of spurious members. The standard deviations estimated for the two clusters being studied are shown in Fig. 7.
Figure 7: Estimated standard deviations as a function of the sampling radius for the clusters NGC 2323 (squares connected by lines) and NGC 2311 (circles connected by lines). The bars indicate the uncertainties obtained from bootstrapping. 

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4.1 Effectiveness of membership determination
It is not possible in practice to quantify the degree of
correlation between identified and true cluster members,
such as the matching fraction in Fig. 4.
Instead, we can use the concept of effectiveness of
membership determination, which is defined to
be (Wu et al. 2002; Tian et al. 1998)
where p(i) is the membership probability of the ith star and N is the sample size. This index measures the effectiveness of the membership determination by measuring the separation between field and cluster populations in the probability histogram. The higher the index E, the more effective the membership determination. The maximum E value is obtained when there are two perfectly separated populations of stars with membership probabilities p(i)=1 and stars with p(i)=0. Figure 8
Figure 8: Effectiveness of membership determination (see Eq. (8)) as a function of the sampling radius for the open cluster NGC 2323 (open squares connected by solid lines) and for simulations using parameter values corresponding to those obtained for NGC 2323 (dashed lines). 

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4.2 Cluster radius and optimal sampling radius
When using only kinematical criteria, we propose that the sample size can substantially alter the results obtained (the memberships and the remaining properties derived from there). Thus, the strategy of choosing a field large enough to be sure of covering more than the entire cluster must be performed carefully, especially in dense star fields. According to our simulations (Sect. 3), the most robust membership estimation is achieved when . This would seem an obvious result, given that for the cluster is subsampled, whereas for the probability of contamination by field stars is increased. This illustrates that it is important to know the cluster radius reliably before estimating memberships. It is difficult to determine precisely the radius of a cluster because the definition of radius is ambiguous itself, since star clusters have no clearly defined natural boundaries. In this work, we have used the usual definition of , which is the radius of the circle containing all the cluster members. Most ``geometric'' definitions tend to overestimate the true size, especially for irregularly shaped clusters (Schmeja & Klessen 2006). But this is not the main problem, which is instead that the independent estimations of cluster radii available in the literature usually differ significantly. Angular sizes listed in catalogues as Webda^{} were compiled from older references (e.g., Lynga 1987) in which most of the apparent diameters were estimated from visual inspection. According to Webda, for NGC 2323 arcmin, whereas Sharma et al. (2006) estimate arcmin. As mentioned above, it is usual practice to choose a field larger than the apparent area covered by the cluster (taken from the literature) to estimate membership probabilities. However, at least when applying the Sanders' method, assigned members will be spread throughout the selected area because of contamination by field stars. It is probably not coincidental that this is true, for example, for probable members in the Dias catalogue (Dias et al. 2002). How reliable are memberships derived from proper motions? It depends on the ``true'' values. Thus, again, a reliable assessment of membership should use some robust estimation of the radius.
A commonly used procedure for determining (or defining) the cluster radius is based on the analysis of the projected radial density profile. Usually, some particular analytical function (for example, a Kinglike model) is fitted to the density profile and the cluster radius is extracted from this fit. A systematic determination of cluster sizes based on objective and uniform estimations of radial density profiles was performed by Kharchenko et al. (2004). One limitation of this method is the sensitivity of the fit to small variations in the distribution of stars, especially for poorly populated open clusters. The most reliable fits are obtained by using only cluster members, but we are then affected again by the problem of membership determination. As an example, we consider Fig. 9, which
Figure 9: Radial density profiles for the cluster NGC 2323 calculated for the cases arcmin (solid circles) and arcmin (open circles). Lines show the best fits to functions of the form (see Eq. (6)). The solid line represents the case arcmin for which , and the dashed line corresponds to arcmin for which . 

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Obviously, any reliable estimation of the cluster radius ultimately depends on the membership determination. Field star contamination may affect the determination of , and what we have demonstrated in this work is that this contamination can become a significant problem if not taken into consideration. Furthermore, even though cluster and field populations were well separated, the estimated radius would depend on the limit magnitude if, for instance, there was mass segregation. This kind of problems is particularly relevant to the development of automated techniques in which it is necessary to establish objective criteria when determining the size of the sample to be processed. What we propose here is to apply any suggested method to several sample sizes and analyze the behavior obtained. It is difficult to establish simple rules for evaluating this behavior because the results will depend directly on both the membership determination algorithm and the input data. However, for the method considered in this work, which is based on two Gaussian populations, the basic procedure can be outlined as follows:
 1.
 An upper limit to can be estimated by fitting the spatial star density to, for example, a King profile. The estimated tidal radius (or, to be conservative, twice its value) may be considered an upper limit to the optimal sampling radius and would define the range of values to be scanned.
 2.
 For each value, cluster memberships and all the relevant quantities (numbers of cluster stars and field stars, centroids with their standard deviations, effectiveness of membership determination) have to be estimated.
 3.
 The next step is to plot the number of cluster members as a function of the sampling radius . If the membership determination works reasonably well, meaning that it presents little contamination by field stars, then we would observe a behavior similar to that seen in Fig. 2: increases as increases until some point (when ) and then remains approximately constant for higher values (or increases at a much slower rate). In this way, we can estimate the cluster size directly from the data and the membership criteria without making any additional assumptions. The optimal sampling radius at which we achieve the most reliable membership estimation is precisely (Fig. 4)
 4.
 If the parametric model does not adequately describe the real data and/or if the internal noise does not have simple properties, then the behavior of the estimated parameters with respect to would differ from that expected. If this were the case, we should plot the fraction of members versus , where we would identify the optimal sampling radius with the minimum in this plot (Fig. 6). In the absence of more accurate information, this value should correspond to the radius at which the membership classification is the most reliable (for this method in a given astrometric catalogue).
 5.
 Our experience indicates that the properties derived from the Sanders' method tend to exhibit noise and it is not always easy to identify precisely the position of specific features (such as the minimum in the versus plot). Some complementary strategies may be useful in identifying or confirming the optimal sampling radius. First, one can consider the variation in the proper motion standard deviation with radius. The dispersion in the cluster proper motions should exhibit a change of slope at radius close to the optimal sampling radius (Fig. 7). Secondly, the maximal effectiveness of membership determination should be reached around (Fig. 8).
5 Conclusions
We have evaluated the performance of the commonly used Sanders' method (Sanders 1971; Vasilevskis et al. 1958; CabreraCaño & Alfaro 1985) for determining star cluster memberships. In general, the results depend on the radius of the field containing the sampled cluster (the sampling radius, ). The main reason for this dependence is the difference between the assumed Gaussian and the true underlying proper motion distributions. The contamination of cluster members by field stars increases as the sampling radius increases. The rate at which this effect occurs depends on the intrinsic characteristics of the data set. There is a threshold value of above which the identified cluster members are highly contaminated by field stars and the effectiveness of membership determination is relatively small. Thus, care must be taken when applying the Sanders' method (by itself or as part of a more extensive procedure) especially when we do not have reliable information about the true cluster radius and/or when the sampling radius is larger than the cluster radius. If this type of effect is not taken into consideration in automated data analysis then significant biases may arise in the derived cluster parameters. The optimal sampling radius can be estimated by plotting the number of cluster members and/or the fraction of members as a function of the sampling radius. Moreover, this type of analysis can also be used as an objective procedure that can be applied systematically to determine cluster radii.
AcknowledgementsWe thank the referee for his/her comments which improved this paper. We acknowledge financial support from MICINN of Spain through grant AYA200764052 and from Consejería de Educación y Ciencia (Junta de Andalucía) through TIC101 and TIC4075. N.S. is supported by a postdoctoral JAEDoc (CSIC) contract. E.J.A. acknowledges financial support from the Spanish MICINN under the ConsoliderIngenio 2010 Program grant CSD200600070: ``First Science with the GTC''.
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Footnotes
 ... Webda^{}
 http://www.univie.ac.at/webda
All Figures
Figure 1: Proper motion for the stars of a random simulation with , , and (see text for details of the meaning of each of these quantities). Left panel shows the distribution for all the 442 simulated stars. Red circles are the field stars centered on (0,0) with and blue circles are the 200 cluster stars centered on (1,0)with . Right panel is a magnification of the central region in which we have marked with circles the stars whose resulting cluster membership probabilities are higher than 0.5 according to the algorithm used. 

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In the text 
Figure 2: Calculated number of field and cluster stars as a function of the sampling radius in units of the cluster radius, , for simulations with the same set of parameters as Fig. 1. a) Simulation with peaked density profile ( ), assigned members are indicated by squares connected by lines. b) Simulation with flat density profile (), members are indicated by circles connected by lines. Assigned field stars are indicated by vertical bars connected by lines, the length of the bars indicating one standard deviation. The real numbers of simulated stars are shown by dashed lines (cluster) and dotted lines (field). 

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In the text 
Figure 3: Calculated fraction of cluster stars as a function of the sampling radius for the same simulations as in Fig. 2. The real (simulated) values are shown by dashed lines. 

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In the text 
Figure 4: Matching fraction of the cluster (see text) as a function of the sampling radius for the same simulations as in Fig. 2. The error bars are of the order of the symbol sizes but are not shown for clarity. 

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In the text 
Figure 5: Number of cluster stars as a function of the sampling radius in arcmin for the open clusters NGC 2323 (squares connected by lines) and NGC 2311 (circles connected by lines). 

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In the text 
Figure 6: Fraction of cluster stars as a function of the sampling radius for NGC 2323 (squares connected by lines) and NGC 2311 (circles connected by lines). Vertical arrows indicate the optimal sampling radii (see text). 

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In the text 
Figure 7: Estimated standard deviations as a function of the sampling radius for the clusters NGC 2323 (squares connected by lines) and NGC 2311 (circles connected by lines). The bars indicate the uncertainties obtained from bootstrapping. 

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In the text 
Figure 8: Effectiveness of membership determination (see Eq. (8)) as a function of the sampling radius for the open cluster NGC 2323 (open squares connected by solid lines) and for simulations using parameter values corresponding to those obtained for NGC 2323 (dashed lines). 

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In the text 
Figure 9: Radial density profiles for the cluster NGC 2323 calculated for the cases arcmin (solid circles) and arcmin (open circles). Lines show the best fits to functions of the form (see Eq. (6)). The solid line represents the case arcmin for which , and the dashed line corresponds to arcmin for which . 

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In the text 
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